Bertrand MICHEL

Laboratoire de Statistique Theorique et AppliqueeUniversite Pierre et Marie Curie

A Statistical Approach to TopologicalData Analysis

Stochastic Geometry ConferenceNantes, April 6 2016

I - Introduction : Statistics andTopological Data Analysis

Topological data analysis and topological inference

• Geometric inference, algebraic topology tools and computationaltopology have recently witnessed important developments with regardsto data analysis, giving birth to the field of topological data analysis(TDA).

• The aim of TDA is to infer relevant, qualitative and quantitative topolog-ical structures (clusters, holes ...) directly from the data.

• The two popular methods in TDA : Mapper algorithm [Singh et al., 2007]and persistent homology [Edelsbrunner et al., 2002].

• Topological inference methods aim toinfer topological properties of an unknowntopological space X, typically from a pointcloud Xn “close” to X.

[distribution of galaxies]

[3D shape database]

[Sensor Data]

Application fields of TDA methods

Topological data analysis methods can be used:

• For exploratory analysis, visualization:

• For feature extraction in supervised settings (prediction) :

[Chazal et al., 2014b]

[Chazal et al., 2015a]

Non exhaustive list of questions for a statistical approach to TDA :

• proving consistency of TDA methods.

• providing confidence regions for topological features and discussing thesignificance of the estimated topological quantities.

• selecting relevant scales at which the topological phenomenon should beconsidered.

• dealing with outliers and providing robust methods for TDA.

• ...

Statistics and TDAUntil very recently, TDA and topological inference mostly relied on deterministicapproaches. Alternatively, a statistical approach to TDA means that :

• we consider data as generated from an unknown distribution

• the inferred topological features by TDA methods are seen as estimatorsof topological quantities describing an underlying object.

II- Homologyand

Persistent homology

Basic tools for TDA : Offsets and Simplicial Complexes

Point clouds in themselves do not carry any non trivial topological or geometricstructure.

For a point cloud Xn in Rd (or in a metric space),the α-offset of Xn is defined by

Xαn =⋃x∈Xn

B(x, α).

More generally, for any compact set X,

Xα :=⋃x∈X

B(x, α) = d−1X ([0, α])

where the distance function dX to X is

dX(y) = infx∈X‖x− y‖ (in Rd)

General idea: deduce from (Xαn)r>0 some topological and geometric informationof an underlying object.

Basic tools for TDA : Offsets and Simplicial Complexes

Non-discrete sets such as offsets, and also continuous mathematical shapes likecurves, surfaces cannot easily be encoded as finite discrete structures.

A geometric simplicial complex C is a set of simplices such that:

• Any face of a simplex from C is also in C.

• The intersection of any two simplices s1, s2 ∈ C is either a face of boths1 and s2, or empty.

Basic tools for TDA : Offsets and Simplicial ComplexesExamples:

• A simplex [x0, x1, · · · , xk] is in the Cech complex Cechα(Xn) if and only

if⋂kj=0B(xj , α) 6= ∅.

• A simplex [x0, x1, · · · , xk] is in the Rips complex Ripsα(Xn) if and only if‖xj − xj′‖ ≤ α for all j, j′ ∈ {1, . . . , k}.

Can be also defined for a set of points in any metric space or for any compactmetric space.

Rips Cech

Nerve Theorem : the offsets Xαn of a point cloud Xn in Rd are homotopy equiv-alent to the Cech complex Cechα(Xn)

Filtrations of simplicial complexes

Given a point cloud Xn in Rd, we generally define a filtration of (nested simpli-cial) complexes by considering all the possibles scale parameters α : (Cα)α∈A

α

Cα1 Cα2

Homology inference

• Singular homology provides a algebraic description of “holes” in a geo-metric shape (connected components, loops, etc ...)

• Betti number βk is the rank of the k-th homology group.

• Computational Topology : Betti numbers can be computed on simplicialcomplexes.

Homology inference [Niyogi et al., 2008 and 2011] [Balakrishnan et al., 2012] :The Betti number (actually the homotopy type) of Riemannian manifolds withpositive reach can be recovered with high probability from offsets of a sample on(or close to) the manifold.

Persistent homology

Starting from a point cloud Xn, let Filt = (Cα)α∈A be a fitration of nestedsimplicial complexes.

• multiscale information ;

• more stable and more robust ;

• (but does not answer the scale selection problem...)

α

Persistent homology: identification of “persistent” topological features along thefiltration.

Barecodes and Persistence Diagrams

Xn

Barecode

Filtration of simplicialcomplexes Filt(Xn)

Offsets

Barecodes and Persistence Diagrams

Dgm (Filt(Xn))Persistence diagram of the

filtration Filt(Xn) built on Xn.

Xn

Barecode

Filtration of simplicialcomplexes Filt(Xn)

Offsets birth

death

Distance between persistence diagrams and stability

birth

death

∞

0

Multiplicity: 2

Add the diagonal

Dgm1

Dgm2

The bottleneck distance between two diagrams Dgm1 and Dgm2 is

db(Dgm1,Dgm2) = infγ∈Γ

supp∈Dgm1

‖p− γ(p)‖∞

where Γ is the set of all the bijections between Dgm1 and Dgm2 and

‖p− q‖∞ = max(|xp − xq|, |yp − yq|).

Distance between persistence diagrams and stability

birth

death

∞

0

Multiplicity: 2

Add the diagonal

Theorem [Chazal et al., 2012]: For any compact metric spaces (X, ρ) and (Y, ρ′),

db (Dgm(Filt(X)),Dgm(Filt(Y))) ≤ 2 dGH (X,Y) .

Consequently, if X and Y are embedded in the same metric space (M, ρ) then

db (Dgm(Filt(X)),Dgm(Filt(Y))) ≤ 2 dH (X,Y) .

Dgm(Filt(Y))

Dgm(Filt(X))

III - Statisticsand

Persistent homology

Persistence diagram inference [Chazal et al., 2014b]

∞

00

Xn Filt(Xn)

Dgm(Filt(Xn))n points sampled in Xaccording to µ

Filt(X)

∞

00

Dgm(Filt(X))

X

Convergence???

Estimator of Dgm(Filt(K))

(M, ρ) metric spaceX compact set in M. well defined for any

compact metric space[Chazal et al., 2012]

Joint work with F. Chazal, M. Glisse and C. Labruere.

Theorem: For a, b > 0 :

supµ∈P(a,b,M)

E[db(Dgm(Filt(Xµ)),Dgm(Filt(Xn)))

]≤ C

(lnn

n

)1/b

where C only depends on a and b.

Under additional technical hypotheses, for any estimator Dgmn of Dgm(Filt(Xµ)):

lim infn→∞

supµ∈P(a,b,M)

E[db(Dgm(Filt(Xµ)), Dgmn)

]≥ C′n−1/b

where C′ is an absolute constant.

For a, b > 0, µ satisfies the (a, b)-standard assumption on its support Xµ if for anyx ∈ Xµ and any r > 0 :

µ(B(x, r)) ≥ min(arb, 1).

P(a, b,M) : set of all the probability measures satisfying the (a, b)-standard as-sumption on the metric space (M, ρ).

Persistence diagram inference [Chazal et al., 2014a]

Confidence sets for persistence diagrams [Fasy et al., 2014]

P(

Dgm(Filt(K)) ∈ R)≥ 1− α ??

Confidence sets for persistence diagrams [Fasy et al., 2014]

Using the Hausdorff stability, we can define confidence sets for persistence dia-grams:

db (Dgm (Filt(K)) ,Dgm (Filt(Xn))) ≤ dH(K,Xn).

It is sufficient to find cn such that

lim supn→∞

(dH(K,Xn) > cn

)≤ α.

P(

Dgm(Filt(K)) ∈ R)≥ 1− α ??

IV - Robust distance functions forTDA and geometric inference

Standard TDA methods are not robust to outliers

Xr :=⋃x∈X

B(x, r)

= d−1X ([0, r])

where the distancefunction dX to X is

dX(y) = infx∈X‖x− y‖

Standard TDA methods are not robust to outliers

Xr :=⋃x∈X

B(x, r)

= d−1X ([0, r])

where the distancefunction dX to X is

dX(y) = infx∈X‖x− y‖

We would like to consider the sub levels of an alternative distance functionrelated to the sampling measure, which support is X, or close to X.

Robust TDA with an alternative distance function ?

Preliminary distance function to a measure P :Let u ∈]0, 1[ be a positive mass, and P a probability measure on Rd:

δP,u(x) = inf {r > 0 : P (B(x, r)) ≥ u}

supp(P )

x

δP,u(x)

u

δP,u is the smallest distance needed tocapture a mass of at least u.

δP,u is the quantile function at u of the r.v.

‖x−X‖

where X ∼ P .

Distance To Measure [Chazal et al., 2011]

Preliminary distance function to a measure P :Let u ∈]0, 1[ be a positive mass, and P a probability measure on Rd:

δP,u(x) = inf {r > 0 : P (B(x, r)) ≥ u}

supp(P )

x

δP,u(x)

u

Definition: Given a probability measure Pon Rd and m > 0, the distance function tothe measureP (DTM) is defined by

dP,m : x ∈ Rd 7→(

1

m

∫ m

0

δ2P,u(x)du

)1/2

Distance To Measure [Chazal et al., 2011]

Distance To Measure [Chazal et al., 2011]

• Stability under Wassertein perturbations:

‖dP,m − dQ,m‖∞ ≤1√mW2(P,Q)

• The function x 7→ d2P,m(x) is semiconcave, this is ensuring strong reg-ularity properties on the geometry of its sublevel sets.

• Consequently, if P is a probability distribution close to P for Wassersteindistance W2, then the sublevel sets of dP ,m provide a topologicallycorrect approximation of the support of P .

Properties of the DTM :

Let X1, . . . , Xn sample according to P and let Pn be the empirical measure.Then

d2Pn,

kn

(x) =n

k

k∑i=1

||x−X(i)||2

where ||X(1) − x|| ≥ ||X(2) − x|| ≥ · · · ≥ ||X(k) − x| · · · ≥ ||X(n) − x||

Distance to The Empirical Measure (DTEM)

x

k = 8

X(i)

Estimation of the DTM via the empirical DTM

Quantity of interest:d2Pn,

kn

(x)− d2P, kn

(x)

• Observe that

d2P,m(x) =1

m

∫ m

0

F−1x (u)du

where Fx is the cdf of ‖x−X‖2 with X ∼ P .

• The distance to the empirical measure is the empirical counter part ofthe distance to P :

dPn,m(x)2 =1

m

∫ m

0

F−1x,n(u)du

where Fx,n is the cdf of ‖x−X‖2 with X ∼ Pn.

• Finally we get that

d2Pn,

kn

(x)− d2P, kn

(x) =1

m

∫ m

0

{F−1x,n(u)− F−1x (u)

}du

[Chazal et al., 2014b] and [Chazal et al., 2015b]

Estimation of the DTM via the empirical DTM

Quantity of interest:d2Pn,

kn

(x)− d2P, kn

(x)

Two complementary approaches of the problem:

• Asymptotic approach : knn = m is fixed and n tends to infinity.

• Non asymptotic approach : n is fixed, and we want a tight controlover the fluctuations of the empirical DTM, in function of k, whichcan be taken very small.

We do not use Wasserstein stability for either of the two approaches.Wasserstein rates of convergence [Fournier and Guillin, 2013 ;Dereich et al.,2013] do not provide tight rates for the DTM in this context.

[Chazal et al., 2014b] and [Chazal et al., 2015b]

Theorem: Let P be a measure on Rd with compact support. Let D be acompact domain on Rd and m ∈ (0, 1). Assume that there exists an uni-form upper bound ωD on the modulus of continuity for the family (F−1x )x∈Dsatisfying

limu→0

ωD(u) = ωD(0) = 0.

Then√n(d2Pn,m − d

2P,m)converges in distribution to B on D, where B is a

centered Gaussian process with covariance kernel

κ(x, y) =1

m2

∫ F−1x (m)

0

∫ F−1y (m)

0

(P[B(x,

√t) ∩B(y,

√s)]−Fx(t)Fy(s)

)ds dt.

Functional convergence [Chazal et al., 2014b]

Modulus of continuity ωx of F−1x : for any v ∈ (0, 1]

ωx(v) := sup(u,u′)∈[0,1]2, u 6=u′, ‖u−u′‖≤v

|F−1x (u)− F−1x (u′)|.

joint work with F. Chazal, B. Fasy, F. Lecci, A. Rinaldo and L. Wasserman

Theorem: Let x be a fixed observation point in Rd. Assume that ωx :(0, 1] → R+ is an upper bound on the modulus of continuity of F−1x . Letk < n

2 . For any λ > 0:

P(∣∣∣d2Pn, kn (x)− d2

P, kn(x)∣∣∣ ≥ λ) ≤ 2 exp

(−n

8

kn[

ωx(kn

)]2λ2)

+ ...

Assume moreover that ωx(u)/u is a non increasing function, then

E(∣∣∣d2Pn, kn (x)− d2

P, kn(x)∣∣∣) ≤ C√

n

√n

kωx

(k

n

).

Fluctuations of the DTEM [Chazal et al., 2015b]

E(∣∣∣d2Pn, kn (x)− d2

P, kn(x)∣∣∣) ≤ Cn

k

1√n

√k

nωx

(k

n

)renormalization by the mass proportion

parametric rate of convergence

localization at the origin

statistical complexityof the problem

joint work with F. Chazal and P. Massart

Fluctuations of the DTEM [Chazal et al., 2015b]

The quantile function F−1x carries some geometric information.For instance ω(0+) = 0 means that the support of dFx is a closed interval.

Bootstrap and significance of topological features[Chazal et al., 2014b]

Aim : studying the persistent homology of the sub-levels of the DTM andproviding confidence regions.

Two alternative boostrap methods :

• by bootstrapping the DTM

• Bottleneck Bootstrap

Bootstrap and significance of topological features[Chazal et al., 2014b]

Bootstrapping the DTM

For m ∈ (0, 1), define cα by

P(√n||d2P,m − d2Pn,m||∞ > cα

)= α.

Let X∗1 , . . . , X∗n be a sample from Pn, and let P ∗n be the corresponding (boot-

strap) empirical measure.

We consider the bootstrap quantity dP∗n ,m(x) of dPn,m.

The bootstrap estimate cα is defined by

P(√

n||d2Pn,m − d2P∗n ,m

||∞ > cα |X1, . . . , Xn

)= α

where cα can be approximated by Monte Carlo.

Theorem: If F−1x is regular enough, the distance to measure function isHadamard differentiable at P . Consequently, the bootstrap method for theDTM is asymptotically valid.

Bootstrap and significance of topological features[Chazal et al., 2014b]

Dgm : persistence diagram of the sub-levels of dP,m

Dgm : persistence diagram of the sub-levels of dPn,m.Let

Cn =

{E ∈ Diag : db(Dgm, E) ≤ cα√

n

},

where Diag is the set of all the persistence diagrams.

Then,

P(Dgm ∈ Cn) = P(

db(Dgm, Dgm) ≤ cα√n

)≥ P

(‖d2P,m − d2Pn,m‖∞ ≤

cα√n

)

Bootstrapping the DTM

Bootstrap estimate

Bootstrap and significance of topological features[Chazal et al., 2014b]

The Bottleneck Bootstrap

Dgm : persistence diagram of the sub-levels of dP,m

Dgm : persistence diagram of the sub-levels of dPn,m.

Dgm∗

: persistence diagram of the sub-levels of dP∗n ,m.

We directly bootstrap in the set of the persistence diagram by considering the

random quantity db(Dgm∗, Dgm). We define tα by

P(√

ndb(Dgm∗, Dgm) > tα |X1, . . . , Xn

)= α.

The quantile tα can be estimated by Monte Carlo.

Bootstrap and significance of topological features[Chazal et al., 2014b]

In practice, the bottleneck bootstrap can lead to more precise inferences becausein many cases the following stability result is not sharp

db(Dgm,Dgm) ≤ ‖d2P,m − d2Pn,m‖∞.

For both methods we can identify significant features by putting a band of size2cα or 2tα around the diagonal:

Concluding remarks

• TDA methods focus on the topological properties (homology / persistenthomology) of a shape.

• TDA methods can be used

– as an “exploratory method”, in particuar when the point cloud issampled on (close to) a real geometric object

– as a “feature extraction” procedure, next these extracted features canbe used for learning purposes.

• TDA is an emerging field, at the interface maths, computer sciences, statis-tics.

• Many topics about the statistical analysis of TDA

• Applications in many fields of sciences ( medecine, biology, dynamic sys-tems, astronomy, dynamical systems, physics ...)

• TDA methods need to bring together Geometric Inference, ComputationalTopology and Geometry, Statistics and Learning methods.

Thank you !

References[Balakrishnan et al., 2012] Balakrishnan, S., Rinaldo, A., Sheehy, D., Singh, A., and Wasser-

man, L. A.. Minimax rates for homology inference. Journal of Machine Learning Research- Proceedings Track, 22:64–72.

[Bubenik, 2015] Bubenik, P. Statistical topological data analysis using persistence landscapes.Journal of Machine Learning Research, 16:77–102.

[Caillerie and Michel, 2011] Caillerie, C. and Michel, B. (2011). Model selection for simplicialapproximation. Foundations of Computational Mathematics, 11(6):707–731.

[Carlsson, 2009] Carlsson, G. Topology and data. AMS Bulletin, 46(2):255–308.

[Chazal and Lieutier, 2007] Chazal, F. and Lieutier, A. Stability and computation of topolog-ical invariants of solids in {\ Bbb R}ˆ n. Discrete & Computational Geometry, 37(4):601–617.

[Chazal et al., 2012] Chazal, F., de Silva, V., Glisse, M., and Oudot, S. The structure andstability of persistence modules. arXiv preprint arXiv:1207.3674.

[Chazal et al., 2014a] Chazal, F., Glisse, M., Labruere, C., and Michel, B. Convergence ratesfor persistence diagram estimation in topological data analysis. To appear in Journal ofMachine Learning Research.

[Chazal et al., 2014b] Chazal, F., Fasy, B. T., Lecci, F., Michel, B., Rinaldo, A., and Wasser-man, L. (2014a). Robust topological inference: Distance to a measure and kernel distance.ArXiv preprint 1412.7197.

References

References[Chazal et al., 2015a] Chazal, F., Fasy, B. T., Lecci, F., Michel, B., Rinaldo, A., and Wasser-

man, L. Subsampling methods for persistent homology. To appear in Proceedings of the 32st International Conference on Machine Learning (ICML-15).

[Chazal et al., 2015b] Chazal, F., Massart, P., and Michel, B. (2015b). Rates of convergencefor robust geometric inference. ArXiv preprint 1505.07602.

[Dereich et al., 2013] Dereich, S., Scheutzow, M., and Schottstedt, R. (2013). Constructivequantization: Approximation by empirical measures. Ann. Inst. H. Poincare Probab. Statist.,49:1183–1203.

[Edelsbrunner et al., 2002] Edelsbrunner, H., Letscher, D., and Zomorodian, A. (2002). Topo-logical persistence and simplification. Discrete Comput. Geom., 28:511–533.

[Federer, 1959] Federer, H. (1959). Curvature measures. Transactions of the American Math-ematical Society, pages 418–491.

[Fasy et al., 2014] Fasy, B. T., Lecci, F., Rinaldo, A., Wasserman, L., Balakrishnan, S., andSingh, A. Confidence sets for persistence diagrams. The Annals of Statistics, 42(6):2301–2339.

[Fournier and Guillin, 2013] Fournier, N. and Guillin, A. (2013). On the rate of convergencein wasserstein distance of the empirical measure. Probability Theory and Related Fields,pages 1–32.

[Genovese et al., 2012] Genovese, C. R., Perone-Pacifico, M., Verdinelli, I., and Wasserman,L. (2012). Manifold estimation and singular deconvolution under hausdorff loss. Ann.Statist., 40:941–963.

References

References[Massart, 2007] Massart, P. (2007). Concentration Inequalities and Model Selection, volume

Lecture Notes in Mathematics 1896. Springer-Verlag.

[Niyogi et al., 2008] Niyogi, P., Smale, S., and Weinberger, S. (2008). Finding the homologyof submanifolds with high confidence from random samples. Discrete & ComputationalGeometry, 39(1-3):419–441.

[Singh et al., 2007] Singh, G., Memoli, F., and Carlsson, G. E. (2007). Topological methodsfor the analysis of high dimensional data sets and 3d object recognition. In SPBG, pages91–100. Citeseer.

[Singh et al., 2009] Singh, A., Scott, C., and Nowak, R. (2009). Adaptive Hausdorff estimationof density level sets. Ann. Statist., 37(5B):2760–2782.

[Turner et al., 2014] Turner, K., Mileyko, Y., Mukherjee, S. and Harer, J. (2014) Frechetmeans for distributions of persistence diagrams. Discrete & Computational Geometry,52(1):44–70 .

References

Topological invariants

For comparing topological spaces, we consider topological invariants (preservedby homeomorphism) : numbers, groups, polynomials.

How topological spaces can be compared from a topological point of view ?

Topological invariants

Homotopy is weaker than homeomorphism but is preserves many topologicalinvariants.

For comparing topological spaces, we consider topological invariants (preservedby homeomorphism) : numbers, groups, polynomials.

How topological spaces can be compared from a topological point of view ?

• Two continous functions f : X → Y and g : X → Y are homotopicif there exists a continous application H : X × [0, 1] → Y such thatH(·, 0) = f and H(·, 1) = g.

• Two topological spaces X and Y are homotopic if there exists two con-tinous applications f : X → Y and g : Y → X such that

– g ◦ f is homotopic to idX ;

– f ◦ g is homotopic to idY ;

Topological Stability and Regularity

Topological inference : under “regularity assumptions”, topological properties ofX can be recovered from (the off-sets) of a close enough object Y.

Topological Stability and Regularity

Topological inference : under “regularity assumptions”, topological properties ofX can be recovered from (the off-sets) of a close enough object Y.

• The local feature size is a local notion of regularity :For x ∈ X, lfsX(x) := d (x,M(Xc)) .

• Weak feature size and its extensions [Chazal and Lieutier, 2007] (by con-sidering the critical values of dX).

• The global version of the local featuresize is the reach [Federer, 1959] :

κ(X) = infx∈Xc

lfsX(x).

The reach is small if either X is notsmooth or if X is close to being self-intersecting.

Topological Stability and Regularity

Topological inference : under “regularity assumptions”, topological properties ofX can be recovered from (the off-sets) of a close enough object Y.

Theorem [Chazal and Lieutier, 2007]: Let X and Y be two compact sets inRd and let ε > 0 be such that dH(X,Y) < ε, wfs(X) > 2ε and wfs(Y) > 2ε.Then for any 0 < α < 2ε, Xα and Yβ are homotopy equivalent.

Example :

dH(X,Y) = inf {α ≥ 0 | X ⊂ Yα and Y ⊂ Xα}

Topological Stability and Regularity

Topological inference : under “regularity assumptions”, topological properties ofX can be recovered from (the off-sets) of a close enough object Y.

Theorem [Chazal and Lieutier, 2007]: Let X and Y be two compact sets inRd and let ε > 0 be such that dH(X,Y) < ε, wfs(X) > 2ε and wfs(Y) > 2ε.Then for any 0 < α < 2ε, Xα and Yβ are homotopy equivalent.

Example :

dH(X,Y) = inf {α ≥ 0 | X ⊂ Yα and Y ⊂ Xα}

Sampling conditions in Hausdorff metric.

Statistical analysis of homotopy inference can be deduced from support estima-tion of a distribution under the Hausdorff metric.